Refinements of Euler's inequalities in plane, space and n-space

Abstract

We improve Euler’s inequality R ≥ 2r, where R and r are triangle’s circumradius and inradius, respectively, and prove some consequences of it. We also show non-Euclidean version of this result. Next, we improve 3D analogue of Euler’s inequality for tetrahedra R ≥ 3r and discuss recursive way to improve analogues of Euler’s inequality for simplices. We end with some open problems, including possible CEEG (classical Euclidean elementary geometry) proof of Grace-Danielsson’s inequality d2 ≤ (R − 3r)(R + r), where d is the distance between the centers of the insphere and the circumsphere of a tetrahedron.